Sum and Difference Identities Key - Pretty Math · Web viewSum and Difference Identities Key Last...

Sum and Difference Identities Classify each statement as true or false, then explain your reasoning.

1) sin(90) = sin(90) + sin(0) 3) sin(30) + sin(60) = sin(90)

2) cos(90) = cos(90) + cos(0) 4) tan(45) + tan(45) = tan(90)

From these examples, recognize that the distributive property does NOT apply when using the trigonometric functions.

…So what rules do apply?

Sum and Difference Identities

sin( A+B )=sin A cosB+cos A sin Btan(A+B)=tan A+ tanB

1− tan A tan Bsin( A−B)=sin A cosB−cos A sin B

cos (A+B)=cosA cosB−sin A sin Btan(A−B )= tan A−tanB

1+ tan A tan Bcos (A−B )=cos A cosB+sin A sin B

A. Demonstrate Demonstrate that each identity works by substituting the given values for A and B.

Sample: cos (A+B)=cosA cosB−sin A sin B , when A= π3 and B=

π6

Write the identity: cos (A+B)=cosA cos B−sin A sin B

Substitute: cos ( π3 +π6 )=cos

π3 cos

π6−sin

π3 sin

π6

Evaluate: cos (π2 )=( 12 )⋅( √3

2 )−( √32 )⋅( 12 )

Simplify: 0=√34− √3

4

0 = 0

5) cos (A−B )=cos A cosB+sin A sin B , when A= π3 and B=

π6

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Sum and Difference Identities

6) sin( A+B )=sin A cosB+cos A sin B , when A= π3 and B=

π6

7) sin( A−B)=sin A cosB−cos A sin B , when A= π3 and B=

π6

8) tan(A−B )=tan A−tanB1+ tan A tan B , when A= π

3 and B=π6

9) tan(A+B)=tan A+ tanB1− tan A tan B , when A= π

3 and B=π6

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Sum and Difference Identities

B. Find Find the exact values for the trigonometric functions at 105 and 15 by writing the angles as a sum or difference of other special values (such as 60 and 45).

Sample: cos (105° )=cos (60°+45 ° )=cos60 °cos 45 °−sin 60 ° sin 45 °

=12⋅√22

−√32

⋅√22

¿ √24 −√6

4 =√2−√64

10) sin(105 ° )=sin(60°+45 ° )

11) tan(105° )= tan(60°+45 °)

12) cos (15° )=cos (60 °−45 ° )

13) sin(15 ° )=sin(60 °−45° )

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Sum and Difference Identities

14) tan (15° )= tan(60 °−45 ° )

C. Find Use the sum and difference identities to evaluate functions of other rotation angles.

R and S are rotation angles (between 0 and 90) whose terminal sides

intersect the unit circle at (725 ,

2425 ) and (

35 ,

45 ), respectively.

15) Write down the sine, cosine and tangent values for each angle.

sin R = cos R = tan R =

sin S = cos S = tan S =

16) With a calculator in degree mode, solve for the measures of R and S. Then use these values to estimate R+S and RS.R = R+S =

S = RS =

17) Use the sum and difference identities to find the exact values for the trigonometric functions for R+S and RS.sin(R−S )= sin(R+S )=

cos (R−S )= cos (R+S )=

18) How can the answers to #16 be used to check the answers to #17 (using a calculator)?

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(1, 0)

SR

54

53 ,

2524

257 ,

R+S

RS

Sum and Difference Identities

D. Prove Use the sum and difference identities to verify (or prove) other identities.

Sample: Verify that sin(π2−x )=cos x

sin( π2−x ) =sin π2 cos x−cosπ2 sin x (Sum/Difference Identity)

=(1)cos x−(0)sin x (Evaluate/Substitute)=cos x (Simplify)

19) Prove cos (π2−x )=sin x

20) Prove sec(π2−x )=csc x

21) Prove cos (−x )=cos x Hint: Rewrite x as “0 – x”

22) Prove Hint: Rewrite x as “0 – x”

23) Prove tan (−x )=− tan x Hint: Rewrite x as “0 – x”

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Sum and Difference Identities KEYClassify each statement as true or false, then explain your reasoning.

TRUE 1) sin(90) = sin(90) + sin(0) FALSE 3) sin(30) + sin(60) = sin(90)Yes, 1 = 1 + 0 No, ½ + /2 1

FALSE 2) cos(90) = cos(90) + cos(0) FALSE 4) tan(45) + tan(45) = tan(90)No, 0 0 + 1 No, Left side = 2, right side is undefined

From these examples, recognize that the distributive property does NOT apply when using the trigonometric functions.

…So what rules do apply?

Sum and Difference Identities

sin( A+B )=sin A cosB+cos A sin Btan(A+B)=tan A+ tanB

1− tan A tan Bsin( A−B)=sin A cosB−cos A sin B

cos (A+B)=cosA cosB−sin A sin Btan(A−B )= tan A−tanB

1+ tan A tan Bcos (A−B )=cos A cosB+sin A sin B

A. Demonstrate Demonstrate that each identity works by substituting the given values for A and B.

Sample: cos (A+B)=cosA cosB−sin A sin B , when A= π3 and B=

π6

Write the identity: cos (A+B)=cosA cos B−sin A sin B

Substitute: cos ( π3 +π6 )=cos

π3 cos

π6−sin

π3 sin

π6

Evaluate: cos (π2 )=( 12 )⋅( √3

2 )−( √32 )⋅( 12 )

Simplify: 0=√34− √3

4

0 = 0

5) cos (A−B )=cos A cosB+sin A sin B , when A= π3 and B=

π6

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Sum and Difference Identities KEY

6) sin( A+B )=sin A cosB+cos A sin B , when A= π3 and B=

π6

7) sin( A−B)=sin A cosB−cos A sin B , when A= π3 and B=

π6

8) tan(A−B )=tan A−tanB1+ tan A tan B , when A= π

3 and B=π6

9) tan(A+B)=tan A+ tanB1− tan A tan B , when A= π

3 and B=π6

(Both sides of the equation are undefined.)

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Sum and Difference Identities KEY

B. Find Find the exact values for the trigonometric functions at 105 and 15 by writing the angles as a sum or difference of other special values (such as 60 and 45).

Sample: cos (105° )=cos (60°+45 ° )=cos60 °cos 45 °−sin 60 ° sin 45 °

=12⋅√22

−√32

⋅√22

¿ √24 −√6

4 =√2−√64

10) sin(105 ° )=sin(60°+45 ° )

11) tan(105° )= tan(60°+45 °)

12) cos (15° )=cos (60 °−45 ° )

13) sin(15 ° )=sin(60 °−45° )

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Sum and Difference Identities KEY

14) tan (15° )= tan(60 °−45 ° )

C. Find Use the sum and difference identities to evaluate functions of other rotation angles.

(1, 0)

SR

54

53 ,

2524

257 ,

R+S

RS

R and S are rotation angles (between 0 and 90) whose terminal sides

intersect the unit circle at (725 ,

2425 ) and (

35 ,

45 ), respectively.

15) Write down the sine, cosine and tangent values for each angle.

sin R = 24/25 cos R = 7/25 tan R = 24/7

sin S = 4/5 cos S = 3/5 tan S = 4/3

16) With a calculator in degree mode, solve for the measures of R and S. Then use these values to estimate R+S and RS.R = 73.74 R+S = 126.87

S = 53.13 RS = 20.61

17) Use the sum and difference identities to find the exact values for the trigonometric functions for R+S and RS.sin(R−S )= sinR cosS cosR sinS sin(R+S )= sinR cosS + cosR sinS

cos (R−S )= cosR cosS + sinR sinS cos (R+S )= cosR cosS sinR sinS

18) How can the answers to #16 be used to check the answers to #17 (using a calculator)?sin(126.87) 0.8 = 4/5. cos(126.87) -0.6 = -3/5. sin(20.61) 0.352 = 44/125. cos(20.61) 0.936 = 117/125. More exact values could be found using the store key for angles R and S, i.e. sin-1(24/25) ENTER 73.73979529 STO ALPHA R and do the same for S, then use sin(R + S).

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Sum and Difference Identities KEY

D. Prove Use the sum and difference identities to verify (or prove) other identities.

Sample: Verify that sin(π2−x )=cos x

sin( π2−x ) =sin π2 cos x−cosπ2 sin x (Sum/Difference Identity)

=(1)cos x−(0)sin x (Evaluate/Substitute)=cos x (Simplify)

19) Prove cos (π2−x )=sin x

20) Prove sec(π2−x )=csc x

21) Prove cos (−x )=cos x Hint: Rewrite x as “0 – x”

22) Prove Hint: Rewrite x as “0 – x”

23) Prove tan (−x )=− tan x Hint: Rewrite x as “0 – x”

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